Find whether a pattern of pair of numbers will repeat Note: Not sure if I chose the appropriate tag
I have a problem where I have a pair of numbers e.g. (10000, 0).
The pair initializes with the first number being larger than the second (second always initializes to zero).
Both number have a max limit; either number in the pair cannot exceed 65000. When a number tries exceeding its limit, it is reset to 0.
Once the pair is initialized, the number are increased as follows:


*

*everytime first number inc by 1, second number inc by 1

*second number can keep increasing if needed, without first number increasing


As time progresses, the first number increases every 30 seconds. Within a 30 second time interval, the number of times the second number is increased is not fixed. 
Edit: the second number can increase a random number of times within a 30 second period. Its not fixed. So when a new 30 second slot comes about, increase second number by 1. Then pick a random number >= zero, and increase the second number by that amount. To the reader: if it makes easier for you assume that the second number can increased by an amount 0 to 500 (in addition to its increment by 1 due to first number increase).
My question is that whether there is a a way to find whether the numbers the pair was initialized with will ever repeat, i.e. will the pair be ever set to (10000, 0) again ?
 A: With probability $1$ the pair $(10000,0)$ will repeat unless there is something special about the increases of the second number.  The first will repeat with a period $65000 \cdot 30$ seconds and will stay at each value for $30$ seconds.  We might as well define $30$ seconds as a period.  In that case the first number is $10000$ once every $65000$ periods.  If the number of increments of the second number rattles around enough, it will eventually be $0$ in the same period when the first is $10000$.  This can fail if the number of increments of the second is $21$ in the first period then $13$ in every period thereafter.  The second will hit $0$ every $5000$ periods, but never when the first is at $10000$.  If there is reasonable randomness to the second increments you can avoid the periodicity and eventually $(10000,0)$ will repeat.  As I have shown, probability $1$ does not mean the opposite is impossible, just that over a long time the probability of the opposite is as small as you want.
